The rational cohomology ring H*(G), where G is a connected Lie
group, is a free exterior algebra on generators of odd degrees, by
a theorem of Hopf. If G is a complex algebraic group, then the
graded algebra H*(G) is equipped with a decreasing filtration
(the Hodge filtration), and the associated graded is still a (bi-graded)
free exterior algebra, as shown by Deligne.

The talk will present an explicit description of differential operators
and differential forms on a connected complex reductive group G,
which "extend" to a suitable compactification of that group. This will
yield another proof of Deligne's result. Generalizations to homogeneous
spaces will also be discussed.

We define exact functors from certain categories of Harish-Chandra
modules of real classical groups to finite dimensional modules over an
associated affine graded Hecke algebra. We show that these functors map
irreducible spherical representations to irreducible spherical
representations and, moreover that they preserve unitarity. This is joint
work with Peter Trapa.

(Joint work with Paul Mezo.)
Let G+ (resp. G) be the group of real points of a possibly
disconnected real reductive group (resp. of its identity
component). We assume that G+ is generated by
one of its connected components, whose set of real points
G' is assumed to be non-empty. Let K be a maximal compact subgroup of the group of real points of the identity component of this algebraic group. We
characterize the space of maps &pi &rarr tr(&pi(f)), where &pi is an irreducible representation of G+ and f varies over the space of smooth
compactly supported functions on G' which are left and right K-finite. This work is motivated by applications to the twisted Arthur-Selberg twisted
trace formula.

This is a joint work with Jorge Vargas. We consider cases of
discrete series for a real reductive group whose restriction to a
reductive subgroup is admissible, and for which the multiplicities can be
computed by a formula of Kostant-Blattner type.

Over the years, the local theta correspondence for reductive dual
pairs of subgroups of the symplectic group has repeatedly provided
insight into representation theory of reductive groups and the theory
of automorphic forms. The most heavily studied example of such a pair
is (O(p, q), SL(2,R)), which provides the setting for the
construction of modular forms, including the classical Jacobi theta
functions, the η function, Hecke's construction of modular forms
associated to quadratic grossencharaktere, and Maass's wave forms, as
well as an analysis of Huyghens' Principle and the Bochner periodicity
relations. This talk will discuss a variant of this pair, involving
the indefinite orthogonal groups O(p, q) and the Lie superalgebra
osp(2, 2). Among the motivations for this study is the close
connection of this pair with Maxwell's equations. Many aspects of the
correspondence for (O(p, q), SL(2,R)) find analogs for (O(p, q),
osp(2, 2)), but in some ways this correspondence is more natural. The
Casimir operator for osp(2, 2) plays a key role. This is a report on
the Yale Ph.D. thesis of Dan Lu.

Minimal representations are the smallest infinite dimensional
unitary representations. The Weil representation for the metaplectic group,
which plays a prominent role in number theory, is a classic example.

We may consider that minimal representations (from the viewpoint of groups)
as "maximal symmetries (from the viewpoint of representation spaces)",
and thus propose to use minimal representations as guiding principles
to find new interactions with other fields of mathematics.

Highlighting geometric analysis on minimal representations
of O(p,q), I plan to discuss conservative quantities of ultrahyperbolic
equations, the generalization of the Fourier-Hankel transform on the
L^2-model, and its deformation.

This is a continuation of my previous work on the original CDSW
conjecture. Let g be a simple Lie algebra with an involution
&sigma and let k (resp. p) be the +1 (resp. -1)
eigenspace of &sigma . We assume that p is an
irreducible k-module. In this setting, we give a generalization
of the original CDSW conjecture. The original conjecture and a brief
outline of its proof will be recalled in the talk. Even though the
techniques used to prove the generalization are similar to the
previous work, there are added topological complications to take care
of.

George Lusztig, From groups to symmetric spaces

A symmetric space is a reductive group with an involution.
Any reductive group G gives rise to a symmetric space (G2,i)
where i(a,b)=(b,a). Thus symmetric spaces can be viewed as
generalization of reductive groups. It has happened many times
that a property/construction for reductive group is a special
case of a property/construction which makes sense for any
symmetric space. In this lecture we will review a number of such
properties/constructions.

We discuss some necessary conditions and sufficients conditions on the
existence problem of the homomorphisms between scalar generalized Verma
modules.
In particular, for complexified minimal parabolic subalgebras of real forms,
we determine such homomorphisms between those with regular infinitesimal
characters except for so(m,n) with m+n odd and sp(m,m).

We characterize rationally smooth K-orbit closures in the
flag variety for SU*(2n) via a pattern avoidance criterion on the
involutions parametrizing them, showing along the way that the
singular locus of any such orbit closure coincides with the rational
singular locus.

Restrictions of zonal spherical functions and Heckman-Opdam's
hypergeometric functions on one-dimensional singular lines
though the origin satisfy interesting ordinary differential
equations. By a unifying study of ordinary differential
equations on a Riemann sphere we have a global structure of
their solutions and for example we have a new proof of
Gindikin-Karpelevic formula of c-functions and the Gauss
summation formula of Heckman-Opdam's hypergeometric functions.

Alessandra Pantano, Complementary Series of Split Real Groups

he aim of this talk is to present some recent progress on the determination of
the unitarizable minimal principal series of split real groups.

In particular, we explore the relation between genuine complementary series of
the metaplectic group Mp(2n), non-spherical complementary series of the split
orthogonal group SO(n+1,n) and the spherical complementary series of certain
other split orthogonal groups.

This is joint work with Annegret Paul and Susana Salamanca-Riba.

Gordan Savin, Parameterizing representations of non-linear groups.

While it is possible to write parameters of representations of linear groups
in a (more or less) canonical fashion, parametrerizations for non-linear
groups depend on certain choices. For example, one can parameterize
representations of the two-fold cover of SL(2) using the dual pair (SL(2),
O(3)) but the transfer of representations from SL(2) to O(3) depends on the
choice of oscillator representation or, equivalently, on the choice of
additive character. In this talk I will describe an approach to this problem
for two-fold covers of simply connected Chevalley groups, and an application
to dual pair correspondences arising from small representations of groups of
type Bn.

A minor extension of M. Saito's theory of mixed Hodge
modules can be used to construct canonical filtrations on Harish-Chandra
modules. They have remarkable formal properties, but are very difficult to
compute explicitly. A key step is to determine a type of b-function. This is
joint work with Kari Vilonen.

Michèle Vergne, &nbsp Hyperplane arrangements and Blattner formula

With Baldoni, Beck, and Cochet, we showed that an efficient way
of computing vector partition functions for classical root systems is
by computing residues of rational functions defined on the complement of
the corresponding arrangement of hyperplanes.
I will discuss applications of this method to Blattner formula for
discrete series of the group U(p,q).

David Vogan, &nbsp Signatures of Hermitian forms and unitary representations

Suppose G is a real reductive Lie group. According to Gelfand's
philosophy of abstract harmonic analysis, the most basic question in
the representation theory of G is the classification of irreducible
unitary representations. One way to approach this question is first
to classify the irreducible hermitian representations: those
admitting a nondegenerate but possibly indefinite invariant hermitian
form. (As evidence that this is a promising path, notice that in the
two classical limiting examples -- when G is compact, and when G is
abelian -- a hermitian representation is automatically unitary.)
Knapp and Zuckerman gave a complete classification of irreducible
hermitian representations in a 1977 paper. What remains is to decide
which of these representations are unitary.

I will describe joint work with Jeffrey Adams, Marc van Leeuwen,
Peter Trapa, and Wai Ling Yee on formulating an algorithm to
calculate the signature of the invariant Hermitian form on any
irreducible Hermitian representation (and so, in particular, to
determine whether the representation is unitary). The formulation of
the algorithm is more or less complete; our proof that the algorithm
is correct still has some missing steps. The main ingredient is the
Beilinson-Bernstein proof of Jantzen's conjecture for Harish-Chandra
modules; the main thing to be computed is Kazhdan-Lusztig polynomials
for real groups.

Nolan Wallach, &nbsp Variations on a theme of Rino Sanchez

In his (unpublished) thesis (2003), Rino Sanchez gave a purely algebraic
construction (and proof of unitarity) of an important representation of the
two-fold cover of SL(3,R) originally constructed analytically by
Torasso in a 1983 paper. We will give a description of Rino's method and give
some extensions.

Chen-Bo Zhu, Archimedean multiplicity one theorems

Let (G, G') be one of the following pairs of classical groups:
(GL(n,R), GL(n-1,R)), (GL(n,C), GL(n-1,C)), (O(p,q),
O(p,q-1) ), (O(n,C), O(n-1,C)),
(U(p,q), U(p,q-1)). We consider the class of irreducible admissible smooth
Frechet representations of moderate growth, for G and G'
respectively. The multiplicity one theorems in the title assert that
any representation of G' in this class occurs (as a quotient) with
multiplicity at most one in any representation of G in the same
class. The talk represents joint work with Binyong Sun. For general
linear groups, this is also in dependently due to A. Aizenbud and
D. Gourevitch.